The Two Special Triangles

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 25 of 46.

Two triangles from geometry give exact values for 30°, 45° and 60°. Cut a square along its diagonal: each half has angles 45°, 45°, 90° and, for legs of 1, a hypotenuse of √2. Cut an equilateral triangle of side 2 down the middle: each half has angles 30°, 60°, 90° and sides 1, √3 and 2, the 1 opposite the 30°.

$Half a square: legs 1 and 1, hypotenuse √2, two 45° angles$

$Half an equilateral triangle: sides 1, √3 and 2, angles 30° and 60°$

Read SOH CAH TOA off them and you get the table below. Two entries were tidied: 1/√2 was written √2/2 and 1/√3 was written √3/3, by multiplying top and bottom by the root.

angle	sin	cos	tan
$30^{\circ}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45^{\circ}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^{\circ}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

With these values a side comes out exact, with a root in it, and no rounding. If you forget one, sketch the triangle and read it off.

Challenge

Triangle ABC has a right angle at C and a 60° angle at A. The hypotenuse is 8. Find the side x opposite the 60° angle exactly, with a root in it. Then press Check.

Try it yourself

$Use the figure to find x$